1. ## Irreducible Polynomials:

(1):
Find all irreducible polynomials of the form $\displaystyle x^2 + ax +b$, where a,b belong to the field $\displaystyle \mathbb{F}_3$ with 3 elements.
Show explicitly that $\displaystyle \mathbb{F}_3(x)/(x^2 + x + 2)$ is a field by computing its multiplicative monoid.
Identify [$\displaystyle \mathbb{F}_3(x)/(x^2 + x + 2)$]* as an abstract group.

Find all irreducible polynomials of the form $\displaystyle x^2 + ax +b$, where a,b belong to the field $\displaystyle \mathbb{F}_3$ with 3 elements.
Just list all of them $\displaystyle x^2 + x+1,x^2+x+2,...$ there are only $\displaystyle 9$ or them. Now, just check which ones have zeros (there are only three zeros to check). And this tells you which are irreducible and which are not.
Show explicitly that $\displaystyle \mathbb{F}_3(x)/(x^2 + x + 2)$ is a field by computing its multiplicative monoid.
Identify [$\displaystyle \mathbb{F}_3(x)/(x^2 + x + 2)$]* as an abstract group.
Any element in $\displaystyle \mathbb{F}_3[x]/(x^2+x+2)$ has form $\displaystyle a+bx+(x^2+x+2)$. Now show that these elements form a field.